Listen to Robert Emmerich introduce The Big Apple, a hit song from 1937. Music written by Bob and performed by Tommy Dorsey's Clambake Seven with Bob on piano. Lyrics written by Buddy Bernier and sung by Edythe Wright. Audio provided by Dorothy Emmerich.

"Shit flows/rolls/runs downhill” means that difficulties flow from the top of an organization to the bottom, and those on the bottom will get the “shit.” The saying has been used in the military and in hospitals since at least 1970.

“Shit rolls down hill” was cited in print since 1970 and “Shit runs downhill” from 1971. The original proverbial saying is that “water runs downhill.”

Google BooksThe Third Face of War
By Gene Schulze
Austin, TX: Pemberton Press
1970
Pg. 274:
“If he wanted to go from top . . . echelon . . . trust, travel, down to dealing with the enemy- Shit rolls down hill.”

Google BooksMaledicta
Volume 4
1980
Pg. 168:
From his point of view the infantryman may well consider himself lower than shit— he catches it from the enemy in combat and from his own Army in garrison, proving the Old Army adage that “shit rolls downhill.”

Google BooksMother Jones
Volume 5
1980
Pg. 38:
The other, by a professor of medicine three years later to an uppity student: “Shit flows downhill and don’t forget that you’re on the bottom.”

Urban Dictionaryshit rolls down hill
when you have to deal with a problem that someone above you (your boss, parents, teachers, etc) caused.
don’t forget....you can throw shit up hill cause it will roll right back down, but heavier.
I gotta do your job cause you fucked up?
Man, shit keeps rolling down hill at this crappy job.
by metamorphosis May 15, 2004